# Questions tagged [bivariate-distributions]

For questions on bivariate distributions, the combined probability distribution of two randomly different variables.

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### Marginal density equal to zero everywhere.

I've been working on a two part question on bivariate transformations and marginal densities but am having difficulty finding where I have made a mistake as the final answer for the marginal density ...
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### Prove Independence of a Bivariate Normal Distribution

Two variables $X,Y$ are bivariate normally distributed. We know that $Var(X)=Var(Y)$. Show that the two random variables $X+Y$ and $X-Y$ are independent. I'm feeling pretty stumped by this question, ...
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### Solution to bivariate normally distributed problem

Is anyone able to solve the following integral with a normal bivariate PDF, or at least to clarify if there exist a closed form solution? Thanks in advance. \begin{equation} \frac{1}{2\pi} \int_{-\...
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Let $X$ and $Y$ be random variables with probability density $f(x, y)=12xy(1-y)$ if $0<x<1, 0<y<1$. Find the joint probability density of $U=XY^2$ and $V=Y$. Find the marginal density of $... 0 votes 0 answers 36 views ### Find marginal probability using bivariate normal I'm given two variables$\begin{bmatrix} X_{1}\\{X_2} \end{bmatrix} \sim N_2({\mu}=\begin{bmatrix} 2\\-5\end{bmatrix},\Sigma = \begin{bmatrix} 1&{-0.5}\\{-0.5}&4 \end{bmatrix})$How would I ... 0 votes 0 answers 14 views ### Trinomial and bivariate normal distribution - checking answers. please could someone check through my answers for this question. Thanks! X,Y,Z follow a trinomial distribution with success probabilities$p_X, p_Y, p_Z > 0$and$p_X + p_Y + p_Z = 1$and sample ... 0 votes 1 answer 184 views ### Expected Value of X,Y from Covariance Matrix Let$(X,Y)$be a bivariate random variable with a Gaussian distribution on$\mathbb{R}^2$, mean zero and variance-covariance matrix: $$C=\begin{pmatrix} 0.42 & -0.42\\-0.42 & 0.42\end{pmatrix}... 0 votes 1 answer 75 views ### Does Covariance equal 0 Let U,V be a bivariate random variable with a continuous distribution and f_{U,V} is the joint density of (U,V). Suppose that f_{U,V}(−u,v)=f_{U,V}(u,v) for all u,v∈\mathbb{R}, then cov(U,V)... 2 votes 0 answers 53 views ### Expectation of |x^{2} - y^{2}| when x and y are bivariate normal I was asked the following question by my professor. Suppose x and y are jointly normal variables. We further suppose the marginal distribution of x and y are N(0, 1) and N(0, 2) ... -1 votes 1 answer 112 views ### Finding max of var(a^TY). Let X=(X_1,X_2,X_3)^T be a multivariate random variable with the standard Gaussian distribution on \mathbb{R}^3. Define the multivariate random variable Y=(Y_1,Y_2,Y_3)^T by$$\begin{pmatrix} ... 2 votes 1 answer 131 views ### Deriving the joint probability distribution of a transformed rv Suppose that A and B are 2 r.v with joint probability distribution given by$f_{AB}(a,b) = -\frac{1}{2}(ln(a)+ln(b))$, if$0 \lt a \lt 1$and$0 \lt b \lt 1$, and 0 otherwise. Define$X = A+B$and$Y =... 1 vote
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Let $A=(M,N)^T$ be a bivariate random variable with joint density defined by $$f_{M,N}(m,n) = \frac{3}{2 \pi} \sqrt{m^2+n^2}$$ if $m^2+n^2<1$ and $0$ otherwise. Let $B = (F,G)^T$ be given by $$\... 0 votes 0 answers 57 views ### Absolute value of expected value Given (A_1,A_2) to be the bivariate random variable in \mathbb{R}^2 with mean 0 and cov(A_1,A_2) = -1.05 and Var(Z_1) = Var(Z_2) = 1.05. How do I find the expected value of E(|A_1A_2|)? ... 0 votes 0 answers 87 views ### Finding covariance matrix from a 3-variate RV Let M be a 3-variate standard Gaussian distribution on \mathbb{R}^3 given by M=(M_1,M_2,M_3)^T .Let V=(V_1,V_2)^T be defined by$$V= \begin{pmatrix}1.1 & M_3\\0 & 1.1\end{pmatrix}\begin{... 1 vote
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### Finding the expected value of a bivariate gaussian distribution

Suppose that $(A,B)$ have a standard Gaussian distribution on $\mathbb{R}^2$. How do I find the expected value for $\mathbb{E}[max(3.9A+B,A+3.9B)]$? I know that A and B follow the standard normal ...
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### Proof that $\mathbb{E}\bigl[1-\Phi(x_1 - \theta_1) \mid \theta_2 \leq x_2-σx_1+σθ_1\bigr]$ is increasing in $\sigma$

Numerically it appears that the following function is increasing (or at least non-decreasing) in $σ$, $$f(x_1,x_2;σ)=∫_{-∞}^{∞}∫_{-∞}^{x_2-σx_1+σθ_1}[1-Φ(x_1-θ_1)]φ(θ_1)φ(θ_2)dθ_2dθ_1$$ where $\phi$ ...
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### Finding $P(X \leq a | Y = k)$ where $X\sim\operatorname{Exp}(\lambda)$ and $Y = [X]$ where $[\cdot]$ rounds up to the nearest integer.

$Y = [X]$ where $[\cdot]$ rounds up to the nearest integer. It's given that $X \sim \operatorname{Exp}(\lambda)$. The first part asks me to show that $Y \sim \operatorname{Geo}(1-e^{-\lambda})$. The ...
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### Guessing a function that maximises the variance?

I'm doing a question that discusses a six-sided fair die being rolled and then discussing the numbers that appear on the top and on the side facing you. The rest of the question discusses the ...
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### In search of a bivariate distribution with nicely behaved integrals over a surface

I am looking for a joint distribution with infinite support on $R^2$, where $P(Y>max(c,aX+b))$, where $a$,$b$ and $c$ are real numbers, has a closed-form solution (without integrals). I tried many ...
I have a random process, $$Y_i=Bf_i+W_i,\quad i=0,\ldots,N$$ The RVs $B,W_1,\ldots,W_n$ are i.i.d., with $B\thicksim\mathcal{N}(0,\sigma_b^2)$ and $W_i\thicksim\mathcal{N}(0,\sigma_w^2)$. The ...